Some Solitons on Homogeneous Almost $\alpha$-Cosymplectic $3$-Manifolds and Harmonic Manifolds

Ricci solitons are the self similar solutions of the Ricci flow. The concept of Ricci flow was first introduced by Hamilton [8] in (1982), motivated by the work of Eells and Sampson [7] on harmonic map and the flow was given by the equation

where $S$ is the Ricci tensor.
Ricci solitons are the generalizations of the Einstein metrics and are the solutions of the equation

where $Ric(X,Y)=S(X,Y)$ is the Ricci curvature tensor, $\mathfrak{L}_{X}$ is the Lie derivative along the direction of the vector field $X$ and $\lambda$ is a real constant. The soliton is said to be shrinking if $\lambda>0$, steady if $\lambda=0$ and expanding if $\lambda<0$.
Tashiro [17] proved very important result for complete Einstein manifolds admitting Ricci solitons.

The Einstein solitons are the generalization of the Ricci solitons, was first introduced by Catino and Mazzieri [4] in (2016). They are the solutions of the equation

where, Ricci tensor $S(X,Y)=g(X,QY)$, $Q$ being the Ricci operator, $r$ is the scalar curvature, $\lambda\in\mathbb{R}$ is a constant and $V$ is known as potential vector field.
Einstein solitons are the self-similar solutions of the Einstein flow,

It is said to be steady if $\lambda=0$, shrinking if $\lambda>0$ and expanding if $\lambda<0$.

In order to define contact metric manifolds, we need the concept of Reeb vector field.
Reeb vector field [3]: A global vector field $\xi$ on a contact manifold $M^{2n+1}$, equipped with a global $1$-form $\eta$, is called Reeb vector field or characteristic vector field, if any vector field $X$ satisfies $\eta(\xi)=1$ and $d\eta(X,\xi)=0$.

Almost contact manifold [3]: Let $M$ be a Riemannian manifold of dimension $(2n+1)$, $n\geq 1$. $M^{2n+1}$ is said to have an almost contact structure $(\varphi,\xi,\eta)$, if there exists a $(1,1)$-tensor $\varphi$, a global vector field $\xi$ and a $1$-form $\eta$ such that

for any vector field $X$ on $M$, where $\xi$ is the Reeb vector field. The manifold $M$ equipped with the structure $(\varphi,\xi,\eta)$ is called an almost contact manifold.

Almost contact metric manifold [3]: A Riemannian metric $g$ is said to be compatible with an almost contact structure $(\varphi,\xi,\eta)$, if

holds for any $X,Y\in\chi(M)$ and $(M,\varphi,\xi,\eta,g)$ is called an almost contact metric manifold.

Normal almost contact metric manifold [3]: An almost contact metric manifold is said to be normal, if for any $X,Y\in\chi(M)$ the tensor field $N=[\varphi,\varphi]+2d\eta\otimes\xi$ vanishes everywhere on the manifold, where $[\varphi,\varphi]$ is the Nijenhuis tensor of $\varphi$.

Homogeneous almost contact metric manifold [10]: An almost contact metric manifold $(M,\varphi,\xi,\eta,g)$ is said to be homogeneous, if there exists a connected Lie group $G$ of isometries acting transitively on $M$ leaving $\eta$ invariant.

A $(2n+1)$-dimensional manifold is said to be a cosymplectic manifold [11], if it admits a closed, 1-form $\eta$ and $2$-form $\Phi$ such that $\eta\wedge\Phi^{n}$ is a volume element, where $\Phi(X,Y)=g(\varphi X,Y)$ is a $2$-form on $M^{2n+1}$.

Almost cosymplectic manifold [11]: If $\eta$ and $\Phi$ are not closed but $\eta\wedge\Phi^{n}$ is a volume form, then the manifold is called almost cosymplectic manifold.

$\alpha$-cosymplectic manifold [14]: An almost cosymplectic manifold is said to be $\alpha$-cosymplectic if $d\eta=0$ and $d\Phi=2\alpha\eta\wedge\Phi$ for some constant $\alpha$.

Almost $\alpha$-cosymplectic manifold [11]: An almost $\alpha$-cosymplectic manifold is defined as an almost contact metric manifold with $d\eta=0$ and $d\Phi=2\alpha\eta\wedge\Phi$, for any constant $\alpha$. In particular, the almost $\alpha$-cosymplectic manifold is

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  almost $\alpha$-Kenmotsu if $\alpha\neq 0$,

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  almost cosymplectic if $\alpha=0$,

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  almost Kenmotsu if $\alpha=1$.

Harmonic vector field [16]: A characteristic vector field $\xi$ on an almost $\alpha$-cosymplectic manifold is harmonic if and only if $\xi$ is an eigenvector field of the Ricci operator $Q$.

In this article, we will mainly focus on $3$-dimensional almost $\alpha$-cosymplectic manifold. In what follows, we will be using the following results.

A complete Riemannian manifold $(M^{n},g)$ is said to be harmonic, if for any $p\in M$, the volume density $\omega_{p}(q)=\sqrt{\det(g_{ij}(q))}$ in normal coordinates, centered at any $p\in M$ is a radial function [2]. Thus,

is density of geodesic sphere, is a radial function. It is known that harmonic manifolds are Einstein [2]. They are naturally classified as per the sign of the Ricci constant. Let $r$ be the constant scalar curvature of $M$.

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  If $r=0$, then $M$ is flat, that is $(M,g)=(\mathbb{R}^{n},Can)$ (Lemma 4.5).

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  If $r>0$, then by Bonnet-Myer’s theorem $M$ is compact with finite fundamental group. They are compact rank one symmetric spaces by a well known result of Szabo (cf. [20]).

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  If $r<0$, then $M$ is non-compact harmonic manifold. They are rank one symmetric spaces of non-compact type, if dimension of $M$ is atmost $5$.

The main result in the theory of harmonic spaces is the Lichnerowicz Conjecture: Any simply connected, complete harmonic manifold is either flat or a rank one symmetric space. By the above classification, we see that the conjecture is resolved for compact harmonic manifolds and is open for non-compact harmonic manifolds of dimension $6$. There are counter examples to the conjecture when dimension is atleast $7$, known as the Damek-Ricci spaces or NA spaces. See for more details references in [20].

In the category of non-compact harmonic manifolds, we will be considering simply connected, complete, non-compact harmonic manifolds. It follows that, these spaces don’t have conjugate points (cf. [20]). Hence, by the Cartan-Hadamard theorem,

is a diffeomorphism and every geodesic of $M$ is a line. That is, if $\gamma_{v}:\mathbb{R}\rightarrow M$ is a geodesic of $M$ with $v\in S_{p}M$, $\gamma_{v}^{\prime}(0)=v$, then $d(\gamma_{v}(t),\gamma_{v}(s))=|t-s|.$